Normal Distribution

The probability that a randomly selected value from a standard normal distribution will fall between -1 and 1 is given by the area under the normal curve between these two values. This area can be calculated using the standard normal distribution table or a calculator and is approximately 0.6826.

To find the probability that X will be greater than 60, we first standardize X using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z = (60 - 50) / 10 = 1. Then, we look up the probability that a standard normal random variable will be greater than 1 in the standard normal distribution table or use a calculator. This probability is approximately 0.1587.

To find the probability that a randomly selected value from this distribution will be between 30 and 50, we first standardize the values using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z1 = (30 - 40) / 5 = -2 and Z2 = (50 - 40) / 5 = 2. Then, we look up the probability that a standard normal random variable will be between -2 and 2 in the standard normal distribution table or use a calculator. This probability is approximately 0.6826.

By the Central Limit Theorem, the distribution of sample means will be approximately normal with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. In this case, the mean of the sample means will be 100 grams and the standard deviation of the sample means will be 10 grams / √100 = 1 gram. To find the probability that the average weight of the sample will be between 95 and 105 grams, we standardize the values using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z1 = (95 - 100) / 1 = -5 and Z2 = (105 - 100) / 1 = 5. Then, we look up the probability that a standard normal random variable will be between -5 and 5 in the standard normal distribution table or use a calculator. This probability is approximately 0.9545.

To find the probability that X will be less than -2, we look up the probability that a standard normal random variable will be less than -2 in the standard normal distribution table or use a calculator. This probability is approximately 0.0228.

To find the probability that a randomly selected value from this distribution will be between 80 and 120, we first standardize the values using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z1 = (80 - 100) / 15 = -1.33 and Z2 = (120 - 100) / 15 = 1.33. Then, we look up the probability that a standard normal random variable will be between -1.33 and 1.33 in the standard normal distribution table or use a calculator. This probability is approximately 0.8413.

To find the probability that X will be between 40 and 60, we first standardize the values using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z1 = (40 - 50) / 10 = -1 and Z2 = (60 - 50) / 10 = 1. Then, we look up the probability that a standard normal random variable will be between -1 and 1 in the standard normal distribution table or use a calculator. This probability is approximately 0.6826.

By the Central Limit Theorem, the distribution of sample means will be approximately normal with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. In this case, the mean of the sample means will be 100 grams and the standard deviation of the sample means will be 10 grams / √50 = 1.414 grams. To find the probability that the average weight of the sample will be between 95 and 105 grams, we standardize the values using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z1 = (95 - 100) / 1.414 = -3.53 and Z2 = (105 - 100) / 1.414 = 3.53. Then, we look up the probability that a standard normal random variable will be between -3.53 and 3.53 in the standard normal distribution table or use a calculator. This probability is approximately 0.8413.

To find the probability that X will be greater than 1, we look up the probability that a standard normal random variable will be greater than 1 in the standard normal distribution table or use a calculator. This probability is approximately 0.1587.

To find the probability that a randomly selected value from this distribution will be less than 80, we first standardize the value using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z = (80 - 100) / 15 = -1.33. Then, we look up the probability that a standard normal random variable will be less than -1.33 in the standard normal distribution table or use a calculator. This probability is approximately 0.0228.

To find the probability that X will be less than 40, we first standardize the value using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z = (40 - 50) / 10 = -1. Then, we look up the probability that a standard normal random variable will be less than -1 in the standard normal distribution table or use a calculator. This probability is approximately 0.0228.

By the Central Limit Theorem, the distribution of sample means will be approximately normal with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. In this case, the mean of the sample means will be 100 grams and the standard deviation of the sample means will be 10 grams / √100 = 1 gram. To find the probability that the average weight of the sample will be less than 95 grams, we standardize the value using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z = (95 - 100) / 1 = -5. Then, we look up the probability that a standard normal random variable will be less than -5 in the standard normal distribution table or use a calculator. This probability is approximately 0.0228.

To find the probability that X will be between -1 and 1, we look up the probability that a standard normal random variable will be between -1 and 1 in the standard normal distribution table or use a calculator. This probability is approximately 0.6826.

To find the probability that a randomly selected value from this distribution will be between 70 and 130, we first standardize the values using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z1 = (70 - 100) / 15 = -2 and Z2 = (130 - 100) / 15 = 2. Then, we look up the probability that a standard normal random variable will be between -2 and 2 in the standard normal distribution table or use a calculator. This probability is approximately 0.8413.